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''curvature indicatrix''
 
''curvature indicatrix''
  
A plane curve illustrating the normal curvatures of a surface at a point of this surface. The Dupin indicatrix lies in the tangent plane to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341801.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341802.png" /> and there it is described by the radius vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341803.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341805.png" /> is the normal curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341806.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341807.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341808.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d0341809.png" /> be a parametrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418010.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418011.png" />. One introduces a coordinate system on the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418013.png" />, taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418014.png" /> as the coordinate origin, and the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418016.png" /> as the basis vectors of this coordinate system. The equation of the Dupin indicatrix will then be
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A plane curve illustrating the normal curvatures of a surface at a point of this surface. The Dupin indicatrix lies in the tangent plane to the surface $S$ at the point $P$ and there it is described by the radius vector $r$ of length $1/\sqrt{|K_r|}$, where $K_r$ is the normal curvature of $S$ at $P$ in the direction $r$. Let $\mathbf r=\mathbf r(u,v)$ be a parametrization of $S$ in a neighbourhood of $P$. One introduces a coordinate system on the tangent plane to $S$ at $P$, taking $P$ as the coordinate origin, and the vectors $\mathbf r_u$ and $\mathbf r_v$ as the basis vectors of this coordinate system. The equation of the Dupin indicatrix will then be
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418017.png" /></td> </tr></table>
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$$|Lx^2+2Mxy+Ny^2|=1,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418019.png" /> are the coordinates of a point on the Dupin indicatrix, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418022.png" /> are the coefficients of the second fundamental form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418023.png" /> calculated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418024.png" />. The Dupin indicatrix is: a) an ellipse if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418025.png" /> is an [[Elliptic point|elliptic point]] (a circle if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418026.png" /> is an [[Umbilical point|umbilical point]]); b) a pair of conjugate hyperbolas if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418027.png" /> is a [[Hyperbolic point|hyperbolic point]]; and c) a pair of parallel straight lines if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418028.png" /> is a [[Parabolic point|parabolic point]]. The curve is named after Ch. Dupin (1813), who was the first to use this curve in the study of surfaces.
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where $x$ and $y$ are the coordinates of a point on the Dupin indicatrix, and $L$, $M$ and $N$ are the coefficients of the second fundamental form of $S$ calculated at $P$. The Dupin indicatrix is: a) an ellipse if $P$ is an [[Elliptic point|elliptic point]] (a circle if $P$ is an [[Umbilical point|umbilical point]]); b) a pair of conjugate hyperbolas if $P$ is a [[Hyperbolic point|hyperbolic point]]; and c) a pair of parallel straight lines if $P$ is a [[Parabolic point|parabolic point]]. The curve is named after Ch. Dupin (1813), who was the first to use this curve in the study of surfaces.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d034180a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d034180a.gif" />
  
 
Figure: d034180a
 
Figure: d034180a
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR></table>
 
 
  
  
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The Dupin indicatrix does not exist at a [[Flat point|flat point]].
 
The Dupin indicatrix does not exist at a [[Flat point|flat point]].
  
The Dupin indicatrix at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418029.png" /> can be obtained as the limit of suitably normalized (inter)sections of the surface with planes parallel to the tangent plane of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418030.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034180/d03418031.png" /> which are approaching this plane, see [[#References|[a1]]], p. 370; [[#References|[a2]]], p. 363-365.
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The Dupin indicatrix at $P$ can be obtained as the limit of suitably normalized (inter)sections of the surface with planes parallel to the tangent plane of $S$ at $P$ which are approaching this plane, see [[#References|[a1]]], p. 370; [[#References|[a2]]], p. 363-365.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  (In Russian)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Hsiung,  "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR></table>
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Latest revision as of 12:02, 26 March 2023

curvature indicatrix

A plane curve illustrating the normal curvatures of a surface at a point of this surface. The Dupin indicatrix lies in the tangent plane to the surface $S$ at the point $P$ and there it is described by the radius vector $r$ of length $1/\sqrt{|K_r|}$, where $K_r$ is the normal curvature of $S$ at $P$ in the direction $r$. Let $\mathbf r=\mathbf r(u,v)$ be a parametrization of $S$ in a neighbourhood of $P$. One introduces a coordinate system on the tangent plane to $S$ at $P$, taking $P$ as the coordinate origin, and the vectors $\mathbf r_u$ and $\mathbf r_v$ as the basis vectors of this coordinate system. The equation of the Dupin indicatrix will then be

$$|Lx^2+2Mxy+Ny^2|=1,$$

where $x$ and $y$ are the coordinates of a point on the Dupin indicatrix, and $L$, $M$ and $N$ are the coefficients of the second fundamental form of $S$ calculated at $P$. The Dupin indicatrix is: a) an ellipse if $P$ is an elliptic point (a circle if $P$ is an umbilical point); b) a pair of conjugate hyperbolas if $P$ is a hyperbolic point; and c) a pair of parallel straight lines if $P$ is a parabolic point. The curve is named after Ch. Dupin (1813), who was the first to use this curve in the study of surfaces.

Figure: d034180a


Comments

The Dupin indicatrix does not exist at a flat point.

The Dupin indicatrix at $P$ can be obtained as the limit of suitably normalized (inter)sections of the surface with planes parallel to the tangent plane of $S$ at $P$ which are approaching this plane, see [a1], p. 370; [a2], p. 363-365.

References

[1] V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian)
[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4


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How to Cite This Entry:
Dupin indicatrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dupin_indicatrix&oldid=13694
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article